Total Factor Productivity Growth and Convergence in Petroleum Industry:
An Empirical Analysis Testing for Non-Convexity
Shunsuke Managi1 and Kristiaan Kerstens2
June 2009
Abstract:
While economic theory acknowledges that some features of technology (e.g., indivisibilities, economies of scale and specialization) can fundamentally violate the traditional convexity assumption, almost all empirical studies take the convexity property by faith. In this contribution, we apply two alternative flexible production technologies to measure total factor productivity growth and test the significance of the convexity axiom using a nonparametric test of closeness between unknown distributions. Using unique field level data on petroleum industry, the empirical results reveal significant differences indicating that this production technology is most likely non-convex. Furthermore, we also show the impact of convexity on answering traditional convergence questions in the productivity growth literature.
Key words: Productivity, Luenberger Productivity Indicator, Non-Convexity, Convergence. JEL Classification codes: C430, L710
Acknowledgments: The authors thank James J. Opaluch, Di Jin and Thomas A. Grigalunas for sharing the data in the project of United States Environmental Protection Agency STAR Grant Program (Grant R826610-01). The results and conclusions of this paper do not necessary represent the views of the funding agency.
1 Faculty of Business Administration, Yokohama National University, 79-4, Tokiwadai, Hodogaya-ku, Yokohama 240-8501 Japan. Tel: 81- 45-339-3751, Fax: 81- 45-339-3707, [email protected] (Corresponding author). 2 CNRS-LEM (UMR 8179), IESEG School of Management, 3 rue de la Digue, F-59000 Lille, France.
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1. Introduction
Indivisibility implies that inputs and outputs are not necessary perfectly divisible and also
that scaling up or down the entire production process in infinitesimal fractions may not be
feasible.1 Economies of scale and specialization (implied by the presence of indivisibilities
and other forms of non-convexities in production) entail that a larger per-capita production
increases the extent of the market, facilitates the division of labor, and increases the
efficiency of production. 2
Therefore, for the sake of relevance in both economic theory and associated
empirical analysis one cannot ignore the potential impact of non-convexity, even if this
implies a risk of producing less “elegant” results compared to standard approaches.
However, almost no previous studies have directly tested for the existence of
(non-)convexities in production by rigorous statistical techniques. One possible reason
might be that the widespread perception of convexity as a theoretically sound regularity
property may have discouraged empirical testing. Another possible reason might be that
existing specification tests are unsatisfactory for this purpose, because they may confuse
These economically important features of technology
fundamentally violate convexity of the production possibility set. But, in traditional
empirical analysis (e.g., traditional parametric production analysis, or even nonparametric
production analysis) these features are assumed away by imposing the convexity axiom. In
reality, it is clear that non-convexities in production are sufficiently important to explain
behavior in some industries and are critical in the development of the new growth theory
(see, e.g., Romer, 1990, on nonrival inputs). In a similar vein, already McFadden (1978)
recognized that the importance of convexity in production analysis lies in its analytic
convenience rather than its economic realism.
1 Scarf (1986, 1994) stresses the importance of indivisibilities in selecting among technological options. 2 Wibe (1984) shows scale elasticity is generally larger than one for all output levels when reviewing studies
of engineering production functions.
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non-convexities with the effect of small sample error. However, non-convexities in
production play an important role in the theoretical micro-economic literature and have
been studied for decades (see Villar, 1999). For instance, the general equilibrium theory of
non-convex technologies has been thoroughly analyzed (e.g., Joshi, 1997).
In this paper, we apply two alternative flexible production models defined using
nonparametric specifications of technology and test the validity of the non-convexity
assumption in production (NCP). One convexity-free specification of production
technology is the non-convex Free Disposable Hull (FDH) model (initiated by Deprins et
al., 1984). FDH only imposes the assumption of strong or free disposability of both inputs
and outputs. Another more common technology specification adds convexity to these
strong disposability axioms to form a convex nonparametric production model (CP) (see,
e.g., Afriat, 1972 or Färe et al., 1994).
Based on distance functions as representations of technology (and their
interpretation as efficiency indicators (introduced by Farrell, 1957)) computed relative to
both these non-convex and convex nonparametric specifications of technology, we test the
significance of the differences using Li’s (1996) nonparametric test of closeness between
two unknown distributions resulting from independent or dependent observations. If the
true production possibility set is convex, CP and NCP estimators yield approximately the
same results in large data sets, though the NCP model might have a slower rate of
convergence. However, if technology is non-convex, then the CP model offers an
inconsistent approximation and convexity should not be imposed on the specification (see
Simar and Wilson, 2008).
These nonparametric specifications require large data sets on production
technologies to avoid a small sample error problem. Furthermore, to avoid any aggregation
bias the analysis should ideally focus on firm level data with sufficient details on the
production process. Here, we apply this test of convexity to a unique field level data set of
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the petroleum industry in the US Gulf of Mexico over the period 1947 to 1998. Though the
production possibility set of oil and gas development and exploitation is acknowledged to
be non-convex in part of the literature (see, e.g., Devine and Lesso (1972) and further
arguments below), we are unaware of previous economic studies putting this assumption to
an empirical test. Hence, whether the above NCP methodology yields a relevant reference
technology in this industry becomes a most interesting empirical question for testing.
Furthermore, a topic that has received widespread attention with the appearance of
endogenous growth theories is the question of convergence in productivity levels (see Islam
(2003) for a survey). In view of the importance of non-convexities for growth theory
(Romer, 1990), we consider the suggestion by Bernard and Jones (1996) that “future work
on convergence should focus much more carefully on technology”. In particular, we
investigate the issue of convergence/divergence in total factor productivity change using a
recent discrete time Luenberger productivity indicator (Chambers, 2002) while testing for
the significance of the eventual differences between CP and NCP models. The very length
of the observation period provides ample scope to test the impact of the convexity
assumption on the eventual convergence of total factor productivity growth rates.
The choice between non-convexity and convexity in measuring total factor
productivity change relates to the nature of technical change. The NCP model has the
advantage of eventually allowing for local instead of global technical change. While this
distinction between local and global technological change plays a role in some theoretical
work (see, e.g., Atkinson and Stiglitz, 1969, among others), we are unaware of any
empirical work raising this issue. If NCP is the true representation of technology, then
previous empirical work on the convergence issue might not have been reliable.
Anticipating one of the key results, this study only finds convergence for the NCP model.
This contribution is structured as follows. Section 2 reviews the background
literature. Section 3 presents the Luenberger productivity indicator as well as its underlying
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distance functions and the econometric models employed to test for convergence. Section 4
discusses the empirical results and provides the outcomes of the statistical tests. The final
section offers some concluding remarks.
2. Non-Convexity in Production: Literature Review
Interestingly, the literature on non-parametric production analysis (see e.g., Afriat, 1972 or
Varian, 1984) typically uses convexity only as an instrumental regularity property on
technology which is justified by the assumed economic optimization hypotheses. Thus,
convexity is motivated from the perspective of the economic objectives (such as cost
minimization or profit maximization), not as an inherent feature of technology. Similarly,
the parametric approach (see, e.g., Bauer, 1990) sometimes imposes regularity restrictions
on the parameters of cost, revenue and profit functions, but it does not systematically test
for the convexity assumption of technology.
Thereby, the impact of convexity or not of technology on the cost function is often
ignored. While the general property that the cost function is non-decreasing in outputs is
well-known, it seems often forgotten that cost functions estimated on convex (non-convex)
technologies are furthermore convex (non-convex) in the outputs. Jacobsen (1970) was one
of the first to point out that convexity of the cost function in the outputs is due to convexity
of the output set (see proposition 5.2).
Several empirical studies suggest violations of convexity in a wide variety of
industries. Indivisibilities are an obvious feature of real world production settings (see
Scarf 1986, 1994). Phenomena of economies of scale and specialization have equally been
empirically tested in the literature. The empirical evidence of process analysis, which
derives production relations directly from theoretical and practical engineering knowledge,
has found evidence of violations of convexity (see Wibe, 1984). Especially economies of
scale are well documented. For instance, Chenery (1949) studied engineering production
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functions of the pipeline transportation of natural gas and derived a (non-linear) cost
function that exhibits economies of scale. Some evidence of increasing returns has been
reported in, for example, chemical industries, manufacturing of process equipment, air
pollution control equipment, and in biopharmaceutical equipment (e.g., see the survey in
Wibe, 1984). Some other economic analyses documenting these types of violations of
convexity include Yang and Rice (1994) and Borland and Yang (1995).
We provide an analysis of the offshore oil and gas industry which faces substantial
sunk cost investments in development, exploration and knowledge, which are the main
source of non-convexity in production (see Devine and Lesso, 1972, or Frair and Devine,
1975). To enter into some details, once an oil field is found after extensive seismic study
via drilling an exploratory well, other “step-out wells” are drilled to obtain more precise
information on the characteristics of the field (e.g., size being one critical factor). Assuming
entering the production phase is an option (e.g., because of favorable well flow tests), the
above information is exploited to decide on possible locations for the production wells
(specified by map coordinates and depth).
An offshore oil field is developed by drilling directionally from a series of fixed
platforms, whereby drilling and completion costs for a production well are a function of the
length and angle of the hole drilled from the platform to the target. Each platform involves
a huge investment, the precise costs depending on water depth and the number of wells to
be drilled from the platform. Thus, the "platform location problem" is a complex integer
optimization problem aiming to minimize the sum of platform and drilling costs by
determining the number, size, and location of platforms and the allocation of wells to
platforms. Needless to say, small deviations from optimality can generate substantial
inefficiencies. For example, injection wells and other capital equipment for extraction are
sunk and can not readily be changed because of their geometric features.
Further difficult optimization issues at the level of the oil field are, for instance,
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the determination of the optimal production rate for each time period to maximize the total
discounted after-tax cash flow over a specific planning horizon in multilayer oil and gas
fields by exercising control on the number of wells acting during the field's exploitation
period (e.g., Babayev, 1975). Also Neiro and Pinto (2004) argue that the planning and
scheduling of subsystems of the petroleum supply chain (oil field infrastructure, crude oil
supply, refinery operations and product transportation) requires non-convex and nonlinear
mixed-integer optimization models (see also the survey of Durrer and Slater, 1977). In brief,
optimization problems in this sector are hard because of the integer nature of certain
decisions, the nonlinearities and dynamics involved, and the intrinsic uncertainties
surrounding critical parameters (see, e.g., Dempster, et al., 2000).
However, in most of the rather limited economic literature on the oil sector the
issue of convexity seems to be totally ignored. Just to offer one example representative of
similar studies, Cuddington and Moss (2001) estimate average cost functions for additional
petroleum applying error correction models over the period 1967-1990. Using aggregate
data in US, they find the impact of technological change on finding costs for crude oil is
large.
3. A Discrete-Time Luenberger Productivity Indicator: Definitions and
Technology Specifications
Chambers and Pope (1996) define a discrete-time Luenberger productivity indicator in
terms of differences between directional distance functions (see also Chambers, 2002).3
3 Iindicators (indexes) denote productivity measures based on differences (ratios) (see Diewert, 2005).
This is the most general primal productivity indicator currently available, since it is based
upon the directional distance function as a general representation of technology
(Luenberger, 1992; Chambers, Chung, and Färe, 1998). Due to its dual relation with the
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profit function, the latter distance function generalizes the traditional input- or
output-oriented Shephard (1970) distance functions, which are dual to the cost and the
revenue functions, respectively.4
This section discusses a three-step approach to analyze total factor productivity in
the petroleum industry. In the first step, a discrete time Luenberger total factor productivity
indicator is defined. Next, the directional distance functions constituting this indicator are
measured relative to two different technology specifications: one non-convex, the other
convex. In the second step , these two models are statistically compared using a
nonparametric test statistic comparing densities developed by Li (1996). In the third step,
we investigate whether there is
β - and/or σ -convergence in total factor productivity
change for both models using proper econometric models.
Definitions of Technology, Distance Function and Luenberger Productivity Indicator
Using the index set K = {1, …, K} for production units, let
1( ,..., )N Nx x x += ∈ and 1( ,..., )M My y y += ∈ be the vectors of inputs and outputs,
respectively, and define the technology or production possibility set as follows:
{ }t t t t( , ) can produce N MtT x y x y+
+= ∈ (1)
This technology set T consists of all feasible input vectors xt and output vectors yt at time
period t and satisfies certain minimal axioms sufficient to define meaningful distance
functions (see Afriat, 1972). Multi-input, multi-output production technologies and their
boundaries (frontiers) can be characterized by distance or gauge functions, without any
assumption of optimizing behavior on the part of individual observations.5
4 Hence, the more popular Malmquist input- and output-oriented productivity indexes (Caves, Christensen
and Diewert, 1982) based upon these Shephardian distance functions are less general than the Luenberger
productivity indicator.
5 In economics, distance functions are related to the notion of the coefficient of resource utilization (Debreu,
1951) and to efficiency measures (Farrell, 1957). Avoiding these maintained hypotheses may be an advantage,
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Luenberger (1992) generalizes the traditional Shephardian notion of distance
functions by introducing the shortage function that provides a flexible tool capable of
taking account both input contractions and output improvements when measuring
efficiency. Following Chambers et al. (1998), the proportional distance function is defined
as follows:
( ) { }max : ((1- ) (1 ), ; , , )tt t t t t t tD x y x y x y Tδ δ δ= + ∈− (2)
This distance function completely characterizes the technology at period t. Note that this
proportional distance function is a special case of the shortage or directional distance
function. The latter is defined using a general directional vector (-gi, go), whereas the
proportional distance function employs the special case (-gi, go) = (-x, y).
Following Chambers (2002), the Luenberger productivity indicator (total factor
productivity, TFP) in discrete time is defined as follows:
( ) ( )t t t 1 t 1t t t 1 t 1 t t t 1 t 1
1 D (x ,y ) D (x , y ) D (x ,y ) D (x , y ) .2
TFP + ++ + + +
= − + − (5)
Several proportional distance functions are needed to estimate this productivity indicator.
This formulation represents an arithmetic mean between the period t (first difference) and
the period t+1 (second difference) Luenberger indicators, whereby each indicator consists
of a difference between proportional distance functions evaluating observations in period t
and t+1 with respect to a technology in period t respectively period t+1. Taking an
arithmetic mean avoids an arbitrary selection among base years.
This Luenberger productivity indicator can be decomposed into two components:
( ) ( )
t t 1t t t 1 t 1
t 1 t t 1 tt 1 t 1 t 1 t 1 t t t t
D (x ,y ) D (x , y )
1 D (x , y ) D (x , y ) D (x ,y ) D (x ,y ) ,2
TFP ++ +
+ ++ + + +
= −
+ − + − (6)
particularly for micro-level analyses that extend over a long time series with significant uncertainty,
irreversibility and fixed (and/or sunk) costs. In such cases, assumptions of static efficiency of every
production unit in all time periods are likely suspect.
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where the first difference represents efficiency change (EC) and the second term, which is
an arithmetic mean of two differences, represents technological change (TC). While EC
measures changes in the relative position of a production unit relative to the changing
frontier, the TC component provides a local measure of the change in the production
frontier or productivity changes that are due to innovation. To be more precise, TC
measures the arithmetic mean of the productivity change measured from the observation in
period t respectively period t+1 relative to the production frontiers in both periods.
To estimate productivity change over time, four distance functions are needed,
including both within-period and mixed-period distance functions for each field and each
time period. For the mixed-period distance function, we have two years, t and t+1. For
example, 1 1( , )tt tD x y+ + is the value of the proportional distance function for the
input–output vector for period t+1 and technology in period t.
Now we turn to the specification of technology relative to which these distance
functions are estimated. Given the focus on testing the convexity assumption, we look for a
framework allowing defining both a convex and a non-convex representation of
technology.
Non-Convex and Convex Technologies
In principle, distance functions can be estimated using either parametric or
nonparametric specifications of the directional distance function representing technology.
The vast majority of empirical productivity studies seem to employ deterministic,
nonparametric technologies. However, an example of an empirical productivity study using
both nonparametric and parametric technologies is Atkinson, Cornwell and Honerkamp
(2003). While it is common to employ traditional convex nonparametric frontier
technology specifications, in this study we also compute a directional distance function
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relative to a non-convex technology. Therefore, we are able to compare these two
measurements in their effect on the Luenberger productivity indicator and test for the
validity of the convexity axiom. Note that the estimated production frontiers (both CP and
NCP) are based on relative benchmarking among observed units in the marketplace rather
than absolute production possibilities (e.g., engineering estimates). Thus, allowing for any
eventual inefficiency among observations, these frontier estimates provide inner
approximations, i.e., the (non-)convex hull of the underlying unknown true technology.
The construction of nonparametric, deterministic technologies is based on the
minimum extrapolation principle: envelop all observations and extend these using
production axioms about what is considered feasible. Imposing convexity, strong
disposability of inputs and outputs as well as variable returns to scale to obtain a flexible
representation of technology, the proportional distance function at t for CP is defined as:
( ) ( ) ( )D , ; , max( 1 ; 1 ; 1 )},tc t t t t t t t t t t t t t t tx y x y x X y Y e T
δδ δ λ δ λ λ− = − ≥ + ≤ = ∈ (3)
where δ is the maximal proportional amount by which outputs, yt, can be expanded and
inputs, xt, can be reduced simultaneously given the technology, Tt. tY and tX are the
matrices of outputs, ty , and inputs, tx , respectively, while λ is a vector of activity
variables. This CP model involves mathematical programming techniques to estimate the
relative efficiency of all production units relative to best-practice frontiers. Notice that free
disposability is an almost generally accepted assumption in production economics. This
implies that marginal products of inputs, marginal rates of substitution between inputs, and
marginal rates of transformation between outputs are assumed to be non-negative.6
6 Obviously, congestion of production factors can violate this assumption. However, we can interpret
monotonicity as a congestion adjustment to the production set, i.e. the distance functions for monotonized
technologies include both pure technical efficiency and congestion components (see Färe and Grosskopf,
1983). Alternatively, monotonicity can be motivated by the fact that it does not interfere with the
Pareto-Koopmans classification of technical efficiency.
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Next, the NCP can be formulated as follows. The non-convex technology is
obtained from two minimal assumptions: all inputs and outputs are freely (or strongly)
disposable (i.e., monotonicity), and variable returns to scale. Thus, the proportional
distance function at t for the NCP technology set is defined as (see Deprins et al., 1984):
( ) ( ) ( ) { }{ }, ; , max 1 , 1 , 1; 0,1t jn ct t t t t t t t t t t t t t tD x y x y x X y Y e j S
ϖδ δ λ δ λ λ λ− = − ≥ + ≤ = ∈ ∀ ∈
(4)
Notice that the convex monotone hull (3) is obtained by eliminating the binary integer
constraint on the activity variables { }0,1j j Sλ ∈ ∀ ∈ by 0λ ≥ . Actual calculations of the
distance functions under CP and NCP models are provided in the Appendix.
Statistical Analysis of Productivity Growth and Convergence
The estimates of NCP are statistically consistent for a wide range of statistical distributions
(e.g., Park et al., 2000). Consistency also applies for the CP approximation, but only if the
true production set is convex. Hence, if the true production set is convex, CP and NCP
models generally yield approximately the same results in large data sets. However, if the
production set is non-convex, the CP set yields an inconsistent approximation. Therefore, in
large-scale applications, convexity constitutes a potential source of specification error, but
cannot improve the statistical fit.
The convexity axioms in productivity have almost never been exposed to rigorous
empirical testing (see, e.g., Grifell-Tatjé and Kerstens, 2008 for an exception). However, test
procedures for testing the convexity hypothesis in CP model are available. In particular, we
apply the nonparametric Li’s (1996) test to examine the differences in the distribution of
the efficiency scores. Li’s (1996) method tests the closeness of two distributions using
sample distributions based on the kernel density method. The convexity hypothesis is
accepted if there is no statistically significant difference in the efficiency estimates of both
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CP and NCP models. This approach requires large samples; in small samples it can confuse
non-convexities with the small sample error associated with the relaxed model. The
specification tests can test production assumptions only under conditions in which those
assumptions cannot improve the fit of the estimators.
Following the convergence literature, we also report estimates on the
β -convergence and σ -convergence (e.g., Barro and Sala-i-Martin 1992). The
β -convergence notion refers to a tendency of sectors with relatively low initial
productivity levels to grow relatively fast, while the σ -convergence suggests a decreasing
variance of cross-field differences in productivity levels, building upon the proposition that
growth rates tend to decline as sectors approach their steady state.
Despite the popularity of productivity estimates using frontier technologies, notice
that few frontier technology studies have analyzed questions surrounding convergence.
Available frontier-based convergence studies focus most of the times on countries (e.g.,
Henderson and Russell, 2005; Kumar and Russell, 2002; or Kumbhakar and Wang, 2005),
regions (e.g., Salinas-Jiménez, 2003), or sectoral analysis (e.g., Gouyette and Perelman,
1997). Among the exceptions focusing on firm level data within a given industry, one can
notice the articles of Alam and Sickles (2000) on the U.S. airlines industry and Bonaccorsi,
Giuri and Pierotti (2005) analyzing convergence over a relatively long period in the jet
aero-engine industry. Our study focuses on a single industry observed over a long time
period.
4. Empirical Estimation Results
Data used in this analysis are obtained from the U.S. Department of the Interior, Minerials
Management Service (MMS), Gulf of Mexico OCS Regional Office. We have developed a
unique micro (i.e., field) level database using three MMS data files: (1) production data
including well-level monthly oil and gas outputs from 1947 to 1998 (a total of 5,064,843
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observations for 28,946 production wells); (2) borehole data describing drilling activity of
each well from 1947 to 1998 (a total of 37,075 observations); (3) field reserve data
including oil and gas reserve sizes and discovery year of each field from 1947 to 1998 (a
total of 957 observations). Relevant variables were extracted from these data files and
merged by year and field across wells, because the well level is not a good unit to measure
productivity. Due to spillover effects across wells within a given field, the field level is a
more appropriate unit for measuring performance than the well. This explains, for instance,
why companies often engage in joint ventures to exploit a field to internalize the
externalities between wells within the field.
The final data set comprises annual data from 933 fields over a 50-year time
horizon. On average there are 370 fields operating in any particular year, and a total of
18,117 observations. Thus, the database includes field-level annual data over the period
1947 to 1998 for the following variables. Output variables are oil output and gas output.
Input variables are number of exploration and development wells drilled, total drilling
distance of exploration and development wells, number of platforms, water depth, oil
reserve, gas reserve, untreated produced water.
Furthermore, we measure productivity change by looking at relative productivity
across fields of different vintages. In so doing, we are able to separate productivity effects
associated with aging of the field from effects due to differences in the state of technology.
This vintage model differs from the conventional nonparametric model specification in that
the mixed period distance functions compare fields of different vintages for a given field
year, so that the model compares outputs and inputs holding fixed the number of years that
the fields have been operating. Thus, we use cumulative values for inputs and outputs,
because for this nonrenewable industry it is more appropriate to express the production
technology in cumulative terms. For example, for a field, the production at t is determined
by cumulative inputs (e.g., drilling) and outputs (i.e., stock depletion) up to t-1. See the
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Appendix for a more detailed description for this vintage model.
We first examine the convexity hypothesis by comparing the convex and the
convexity-free production models. The TFP in Figure 1 shows how the gross productivity
in offshore oil and gas grows 24.9 and 29.7 % over the study period (a yearly growth rate
of 0.50 and 0.60 %) under the assumption of NCP and CP, respectively. Though the general
trend of these productivities is very similar, their magnitude is clearly different. On average,
these results reflect the productivity-enhancing effects of new technologies becoming
available.
Turning to specific periods, growth is moderate during the 1980s. The latter result
is consistent with common reports of Gulf of Mexico production: it was referred to as the
‘Dead Sea’ in the 1980s. More recent TFP growth has probably resulted because production
has moved to very great water depths in about the last decade or so. While by 1997
production occurred at over a mile deep, by 2001 exploratory wells were being drilled in
nearly 10,000 feet of water. This deep-water production has allowed the discovery of larger
fields.
The lower TFP for NCP might be because the production frontier reacts to the
change in localized productivity change. Atkinson and Stiglitz (1969) analyze the
generation of new technologies and introduced the hypothesis that technological change
can take place only in a limited technical space, defined in terms of both factor intensity
and scale. Technological change is localized because it has limited externalities and affects
only a limited span of the techniques, contained by a given isoquant (see also Stiglitz,
1987; Foray, 1997; Antonelli, 2008). Since technological improvements are in fact
associated with a specific input space, the convexity assumption is likely to overestimate
the true changes of technology. This could explain the higher growth rate under CP.
We also decompose TFP effects into those associated with new innovation (TC)
and catch-up effects (EC). We find significant differences in their sources of TC and EC.
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The result of TC assuming both convexity and non-convexity is provided in Figure 2. TC
increases 15.7 and 27.8 % over the study periods under the assumption of NCP and CP,
respectively. Thus, there are significant increases in TC when imposing the convexity
assumption, while TC is relatively smaller under non-convexity. These disparities can be
explained by the localized nature of technological change (as in the case of TFPs above).
By contrast, we find a significant increase of EC in the non-convex model, while
EC is smaller in convexity model. EC increases 9.2 and 1.9 % over the study periods under
the assumption of NCP and CP, respectively. This might be because the convexity
assumption overestimates the true production frontier shift and therefore EC is obscured in
the case of CP. However, if existing resources are not fully utilized in production initially
(due to technical inefficiencies, variations in capacity utilization, among others), then one
can expect significant scope for variations in EC as revealed by NCP.
Thus, assuming NCP represents the true production frontier, we find TC
contributes less to TFP than EC, while the reverse results hold true for CP.
Non-convex specification can shed light on the local nature of technical change.
Table 1 reports the descriptive statistics over the years and also indicates the number of
observations experiencing progress, regress, and no change. We find more efficient units
and more of these push the frontier up (or down) under NCP than CP for all of TFP, TC,
and EC. Also, focusing on the TC component, we find the number of observations that are
efficient in each period t and that push the frontier forward (or backward) in the period
t/t+1 is larger under NCP than CP. Thus, it is important to focus on individual results in the
non-convex world.
Table 2 shows the results of the Li (1996) test to see whether or not each of the
production models NCP and CP has a different distribution of values for the proportional
distance functions (i.e., “efficiency scores”), as well as the resulting productivity indicator,
and its TC and EC components. In all of these cases, the empirical results indicate that the
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two different distributions of CP and NCP provide statistically significant different patterns.
Therefore, we reject the null hypothesis of distribution closeness between NCP and CP.
To the best of our knowledge, there appear to be no valid theoretical arguments for
assuming a priori that production possibilities are truly convex (see also McFadden, 1978).
In this empirical study, the economically important industry of petroleum exploitation
reveals violations of the convexity hypothesis. Therefore, NCP seems to have a
comparative advantage for analyzing TFP.
Lastly, we investigate the convergence phenomenon in the petroleum industry. In
particular, we estimate the following model consisting of a simple speed-of-convergence
equation ( β -convergence, see Steger, 2000).
( ) 0ln lnit i i iy y eα β∆ = + + ,
where ( )ln it iy∆ shows the indicators of average productivity changes, technological
changes (TC), and efficiency changes (EC) of field i from year 0 to year t for each field i,
0ln iy∆ indicators represents the initial level (field discovered) of the indicators, and ie
represents an error term. The indicators might be resulting from either CP or NCP. A
negative value of β is interpreted as a support for the convergence hypothesis, since it
means that those with lower initial productivity have grown faster over time. Whether or
not we observe convergence might depend crucially on the choice of (non) convexity
assumption.
Table 3 presents the estimation results of convergence in average productivity
changes, technological changes, and efficiency changes for a change in a crucial
technology assumption. The results show that productivity change as well as its both
components converge in NCP over the observation horizon. By contrast, productivity and
efficiency changes do not seem to be converging in CP, though the EC is converging.
Therefore, we have confirmed strong evidence for productivity convergence among
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petroleum fields assuming NCP provides a true measurement. These results might imply
that technology diffusion behind productivity convergence expands opportunities for
secondary firms to catch up to the leading firms. Applying the standard assumption of CP
yields altogether different conclusions.
In Table 4, we test for the eventual existence of σ -convergence for TFP, TC, and
EC in NCP and CP models (see de la Fuente (2002) for a review). Table 4 reports
cross-sectional standard deviations of productivity changes for the two years 1947 and
1998 at the beginning and end of the sample period. The 1998 standard deviations are
greater than those of 1947 for the CP model, while the reverse is true for NCP. Therefore,
assuming NCP, we are able to find the tendency of poor fields to grow faster in a
cross-section bivariate regression of growth rates on initial productivity level and also
tendency of sample dispersion of productivities to diminish. By contrast, no sign of
σ -convergence emerges under the traditional CP.
5. Conclusions
This study raises doubts regarding the ability of traditional convex production functions to
explain the real world phenomenon of observed data. We examined data on petroleum
industry using unique field level data. In this paper, we found that the traditional convex
production model is in trouble and the shapes of the functions are non-convex. In light of
the empirical evidence presented in this study, there is no good reason for considering
convexity of production sets or convexity of input/output sets as generally realistic axioms.
The evidence we have find suggests that non-convexities exist.
It is important to note that the shape of the production function, whether convex or
non-convex, is an important determinant, but not the only determinant, of whether a
production function is an optimal behavior. If the results implying non-convex production
functions are not valid, serious implications for the standard micro-economic theory of the
18
firm follow. The equilibrium of the firm and the existence of competitive markets normally
depend on the convexity of the production functions. It is, therefore, necessary for
researchers to avoid a prior assumption of convexity when the true empirically estimated
functions are non-convex.
19
Appendix: Efficiency Estimation based on (Non-) Convex Production Assumption
When analyzing productive efficiency for extraction of non-renewable resources such as in
the petroleum industry, one faces challenges not met in typical applications to the single
period production of goods and services. For example, production from an oil field at some
point in time depends upon cumulative past production from the field due to depletion
effects, in addition to the technology employed and the attributes of the field (e.g., field
size, porosity, water depth). Holding inputs constant, output from a given field follows a
well-known pattern of an initially increasing output rate, obtaining a peak after some years
of production, then following a long path of declining output (e.g., Pindyck, 1978). This
implies that, for purposes of measuring changes in the productivity, it is inappropriate to
compare contemporaneous levels of output from a newly producing field to a field that has
been producing for ten years or fifty years. Rather, comparisons across fields should be
done holding constant the number of years the fields have been in operation.
Thus, we measure productivity change by looking at relative productivity across
fields of different vintages. By doing so, we separate productivity effects associated with
aging of the field from effects due to differences in the state of technology. The CP
formulation with the vintage model differs from the conventional CP formulation (as
described in, e.g., Färe et al., 1994). Similar to the approach of Managi et al. (2004), our
CP formulation calculates the distance function by solving the following optimization
problem:7
subject to
7 Note that Managi et al. (2004) apply a more traditional Malmquist productivity index method.
' '
' '' ' ' '
,( , ) max
k j
i i i k jc k j k jD
δδ=
λx y
20
( )', '
' '( ) 0
(1 ) , 1,..., ,J k
i k j ikj kjn k j n
k K i jy y n Nλ δ
∈ =
≥ + =∑ ∑
( )', '
' '( ) 0
(1 ) , 1,..., ,J k
i k j ikj kjm k j m
k K i jx x m Mλ δ
∈ =
≤ − =∑ ∑ (A1)
( )
( ) 01,
J k
kjk K i j
λ∈ =
=∑ ∑
).(,...,1),(,0 kJjiKkkj =∈≥λ
where j is the field year, k is the field number, K(i) includes all fields of vintage i (i.e.,
discovered in year i), J(k) is the final year of production for field k, and kjλ is the weight
for field k at field year j.
In a similar manner, our NCP formulation calculates the distance function by
solving the following optimization problem:
subject to
( )', '
' '( ) 0
(1 ) , 1,..., ,J k
i k j ikj kjn k j n
k K i jy y n Nλ ϖ
∈ =
≥ + =∑ ∑
( )', '
' '( ) 0
(1 ) , 1,..., ,J k
i k j ikj kjm k j m
k K i jx x m Mλ ϖ
∈ =
≤ − =∑ ∑ (A2)
{ }( )
( ) 01, 0,1
J k
kj kjk K i j
j Sλ λ∈ =
= ∈ ∀ ∈∑ ∑
).(,...,1),(,0 kJjiKkkj =∈≥λ
We refer to this simple algorithm because it points to an important difference in
the logic that lays behind convex versus non-convex methodologies. The role of the
integrality constraint is indeed essential to identify a dominance relation between observed
production plans. On the one hand, an observation is declared efficient and considered as
lying on the boundary of the reference production set if it is un-dominated. On the other
hand, an observation is declared inefficient, i.e. it lies in the interior of the set, if it is
' '
' '' ' ' '
,( , ) max
k j
i i i k jn c k j k jD
ϖϖ=
λx y
21
dominated by at least one; and in this case the mixed integer program identifies a most
dominating observation that serves as a reference to compute the efficiency degree.
By contrast, the linear programs used in the convex case seek to compute a
distance with respect to the frontier of a convex envelope of the data. While dominance
also plays some role in identifying this envelope, the additional requirement of convexity
induces the possibility that un-dominated observations be declared inefficient because they
do not lie on the convex envelope of the data. Note that the NCP model treats only a
specific and real peer unit, or a collection of such units in the case of a non-unique solution,
as a benchmark at the optimum for the measure of efficiency. Thus, the NCP model
provides a more conservative inner approximation and estimation of the production
possibility set compared to CP.
22
References
Afriat, S. (1972) Efficiency Estimation of Production Functions, International Economic Review 13, 568-598.
Alam I.M.S., and R.C. Sickles (2000) Time Series Analysis of Deregulatory Dynamics and Technical Efficiency: The Case of the US Airline Industry, International Economic Review 41(1), 203-218.
Antonelli, C. (2008) Localised Technological Change, Towards the Economics of Complexity, Routledge.
Atkinson, A.B. and J.E. Stiglitz (1969) A New View of Technological Change, Economic Journal 79(315), 573-578.
Atkinson, S.E., C. Cornwell, and O. Honerkamp (2003) Measuring and Decomposing Productivity Change: Stochastic Distance Function Estimation versus Data Envelopment Analysis, Journal of Business and Economic Statistics 21(2), 284-294.
Babayev, D.A. (1975) Mathematical Models for Optimal Timing of Drilling on Multilayer Oil and Gas Fields, Management Science 21(12), 1361-1369.
Barro, R.J., and X. Sala-i-Martin (1992) Convergence, Journal of Political Economy 100(2), 223-251.
Bauer, P.W. (1990) Recent Developments in the Econometric Estimation of Frontiers, Journal of Econometrics 46(1-2), 39-56.
Bernard, A.B. and C.I. Jones (1996) Technology and Convergence, Economic Journal 106(437), 1037-1044.
Bonaccorsi, A., P. Giuri, and F. Pierotti (2005) Technological Frontiers and Competition in Multi-Technology Sectors, Economics of Innovation and New Technology 14(1-2), 23-42.
Borland, J., and X. Yang (1995) Specialization, Product Development, Evolution of the Institution of the Firm, and Economic Growth, Journal of Evolutionary Economics 5(1), 19-42.
Caves, D.W., L.R. Christensen, and W.E. Diewert (1982) The Economic Theory of Index Numbers and the Measurement of Input, Output, and Productivity, Econometrica 50(6), 1393-1414.
Chambers, R.G. (2002) Exact Nonradial Input, Output, and Productivity Measurement, Economic Theory 20(4), 751–765.
Chambers, R.G., and R.D. Pope (1996) Aggregate Productivity Measures, American Journal of Agricultural Economics 78(5), 1360-1365.
Chambers, R.G., Y. Chung, and R. Färe. (1998) Profit, Directional Distance Functions, and Nerlovian Efficiency, Journal of Optimization Theory and Applications 98(2), 351–364.
Chenery, H.B. (1949) Engineering Production Functions, Quarterly Journal of Economics 63(4), 507-531.
23
Cuddington, J.T., and D.L. Moss, (2001) Technical. Change, Depletion and the US Petroleum Industry: A. New Approach to Measurement and Estimation, American Economic Review 91(4), 1135–48.
de la Fuente, A. (2002) On the Sources of Convergence: A Close Look at the Spanish Regions, European Economic Review, 46(3), 569-599.
Deprins, D., D. Simar, and H. Tulkens (1984) Measuring Labor Efficiency in Post Offices, in: M. Marchand, P. Pestieau and H. Tulkens (eds.) The Performance of Public Enterprises: Concepts and Measurements, North Holland, Amsterdam.
Debreu, G. (1951) The Coefficient of Resource Utilization, Econometrica 19(3), 273–292. Dempster, M.A.H., N. Hicks Pedron, E.A. Medova, J.E. Scott, and A. Sembos (2000)
Planning Logistics Operations in the Oil Industry, Journal of the Operational Research Society 51(11), 1271-1288.
Devine, M.D., and W.G. Lesso (1972) Models for the Minimum Cost Development of Offshore Oil Fields, Management Science 18(8), B378-B387.
Diewert, W.E. (2005) Index Number Theory Using Differences Rather than Ratios, American Journal of Economics and Sociology 64(1), 347-395.
Durrer, E.J., and G.E. Slater (1977) Optimization of Petroleum and Natural Gas Production-A Survey, Management Science 24(1), 35-43.
Färe, R., and S. Grosskopf (1983) Measuring Congestion in Production, Zeitschrift fur Nationalokonomie 43(3), 257-271.
Färe, R., S. Grosskopf, M. Norris, and Z. Zhang. (1994) Productivity Growth, Technical Progress, and Efficiency Change in Industrialized Countries, American Economic Review 84(1), 66–83.
Farrell, M.J. (1957) The Measurement of Productive Efficiency. Journal of the Royal Statistical Society 120(3), 253–281.
Foray, D. (1997) The Dynamic Implications of Increasing Returns: Technological Change and Path Dependent Inefficiency, International Journal of Industrial Organization 15(6), 733-752.
Frair, L., and M. Devine (1975) Economic Optimization of Offshore Petroleum Development, Management Science 21(12), 1370-1379.
Gouyette, C., and S. Perelman (1997) Productivity Convergence in OECD Service Industries, Structural Change and Economic Dynamics 8(3), 279-295.
Grifell-Tatjé, E., and K. Kerstens (2008) Incentive Regulation and the Role of Convexity in Benchmarking Electricity Distribution: Economists versus Engineers, Annals of Public and Cooperative Economics 79(2), 227-248.
Henderson, D.J., and R.R. Russell (2005) Human Capital and Convergence: A Production-Frontier Approach, International Economic Review 46(4), 1167-1205.
Islam, N. (2003) What Have We Learnt from the Convergence Debate?, Journal of
24
Economic Surveys 17(3), 309-362. Jacobsen, S.E. (1970) Production Correspondences, Econometrica 38(5), 754–771. Joshi, S. (1997) Existence in Undiscounted Non-Stationary Non-Convex Multisector
Environments, Journal of Mathematical Economics 28(1), 111-126. Kumar, S., and R.R. Russell (2002) Technological Change, Technological Catch-Up, and
Capital Deepening: Relative Contributions to Growth and Convergence, American Economic Review 92(3), 527-548.
Kumbhakar, S.C., and H. Wang (2005) Estimation of Growth Convergence Using a Stochastic Production Frontier Approach, Economics Letters 88(3), 300-305.
Li, Q. (1996) Nonparametric Testing of Closeness between Two Unknown Distribution Functions, Econometric Reviews 15(3), 261-274.
Luenberger, D.G. (1992) Benefit Functions and Duality, Journal of Mathematical Economics 21(5), 461–481.
Managi, S. J.J. Opaluch, D. Jin, and T.A. Grigalunas (2004) Technological Change and Depletion in Offshore Oil and Gas, Journal of Environmental Economics and Management 47(2), 388–409.
McFadden, D. (1978) Cost, Revenue, and Profit Functions, in: M. Fuss and D. McFadden (eds) Production Economics: A Dual Approach to Theory and Applications, North-Holland, Amsterdam.
Neiro, S.M.S., and J.M. Pinto (2004) A General Modeling Framework for the Operational Planning of Petroleum Supply Chains, Computers and Chemical Engineering 28(6-7), 871–896.
Park, B. U., L. Simar, and C. Weiner (2000) The FDH Estimator for Productivity Efficiency Scores, Econometric Theory 16(6), 855-877.
Pindyck, R.S. (1978) The Optimal Exploration and Production of Nonrenewable Resources, Journal Political Economy 86(5), 841–861.
Romer, P.M. (1990) Are Nonconvexities Important for Understanding Growth?, American Economic Review, 80(2), 97-103.
Salinas-Jiménez, M.M. (2003) Technological Change, Efficiency Gains and Capital Accumulation in Labour Productivity Growth and Convergence: An Application to the Spanish Regions, Applied Economics 35(17), 1839-1851.
Scarf, H.E. (1986) Testing for Optimality in the Absence of Convexity, in: W.P. Heller, R.M. Starr, S.A. Starrett (eds) Social Choice and Public Decision Making: Essays in Honor of Kenneth J. Arrow, Volume I, Cambridge, Cambridge University Press.
Scarf, H.E. (1994) The Allocation of Resources in the Presence of Indivisibilities, Journal of Economic Perspectives 8(4), 111-128.
Simar, L., and P.W. Wilson (2008) Statistical Inference in Nonparametric Frontier Models: Recent Developments and Perspectives, in: Fried, H., Lovell, C.A.K., Schmidt, S. (eds)
25
The Measurement of Productive Efficiency and Productivity Change, Oxford University Press, New York.
Steger, T.M. (2000) Economic Growth with Subsistence Consumption, Journal of Development Economics, 62(2), 343-361.
Stiglitz, J.E. (1987) Learning to Learn, Localized Learning and Technological Progress, in: C.B. McGuire, P. Stoneman (eds) Economic Policy and Technological Performance, Cambridge University Press, Cambridge.
Varian, H.R. (1984) The Nonparametric Approach to Production Analysis, Econometrica 52(3), 579-597.
Villar, A. (1999) Equilibrium and Efficiency in Production Economies, 2nd ed., Springer Verlag, Berlin.
Wibe, S. (1984) Engineering Production Functions: A Survey, Economica 51(204), 401-411.
Yang, X., and R. Rice (1994) An Equilibrium Model Endogenizing the Emergence of a Dual Structure between the Urban and Rural Sectors, Journal of Urban Economics 35(3), 346-368.
26
Figure 1. Productivity Change in Petroleum Industry under Non-Convex and
Convex Assumptions
Figure 2. Technological Change in Petroleum Industry under Non-Convex and
Convex Assumptions
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
1945 1955 1965 1975 1985 1995
TFP (Non-Convex) TFP (Convex)
0
0.05
0.1
0.15
0.2
0.25
0.3
1945 1955 1965 1975 1985 1995
TC (Non-Convex)
TC (Convex)
27
Figure 3. Efficiency Change in Petroleum Industry under Non-Convex and Convex
Assumptions
0
0.02
0.04
0.06
0.08
0.1
1945 1955 1965 1975 1985 1995
EC (Non-Convex)
EC (Convex)
28
Table 1. Luenberger Productivity Indicator: Descriptive Statistics
Non-Convex Production Convex Production
Descriptive Statistics TFP TC EC TFP TC EC
Mean 0.50 0.31 0.18 0.60 0.56 0.04
Standard deviation 0.23 0.27 0.25 0.25 0.31 0.23
Minimum -0.74 -0.69 -0.12 -0.81 -0.77 -0.09
Maximum 1.23 1.22 0.67 1.26 1.31 0.43
Mean # obs. with component > 0 204 211 192 185 195 178
Mean # obs. with component < 0 139 134 156 116 124 140
Mean # obs. with component = 0 27 25 22 69 51 52
Mean # obs. (i) efficient in t and (ii) with component > 0 between t and t+1 - 56 - - 32 -
Mean # obs. (i) efficient in t and (ii) with component < 0 between t and t+1 - 23 - - 17 -
Table 2. Results of the Closeness of Productivity/Efficiency Distribution
Indicators z-test statistics Conclusion
Efficiency Score 3.582*** Reject Null hypotheses
Productivity Change 4.723*** Reject Null hypotheses
Technological Change 5.942*** Reject Null hypotheses
Efficiency Change 3.499*** Reject Null hypotheses
Note: *** Significant at 1 % level.
29
Table 3. Testing Convergence of Productivity Changes
Non-Convex Production
Parameter TFP TC EC β
–0.009** (–2.69)
–0.012** (–2.39)
–0.007*** (–4.71)
R2 0.036 0.027 0.052
Convex Production
Parameter TFP TC EC β
–0.004 (–1.63)
–0.003 (–1.52)
–0.006** (–2.86)
R2 0.012 0.010 0.039
Note.—Values in parentheses are t–values. * Significant at 10% level. ** Significant at 5% level.
*** Significant at 1% level.
Table 4. Testing Convergence of Productivity Changes: σ Convergence
Non-Convex Production
Standard
deviations TFP TC EC
1947 0.283 0.293 0.262
1998 0.256 0.243 0.250
Convex Production
Standard
deviations TFP TC EC
1947 0.243 0.302 0.224
1998 0.256 0.312 0.239